Advertisement

Zero Dynamics and Stabilization for Linear DAEs

  • Thomas Berger
Conference paper
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the asymptotic stability of the zero dynamics and stabilizability. To this end, the concepts of autonomous zero dynamics, transmission zeros, right-invertibility, stabilizability in the behavioral sense and detectability in the behavioral sense are introduced and algebraic characterizations are derived. It is then proved, for the class of right-invertible systems with autonomous zero dynamics, that asymptotic stability of the zero dynamics is equivalent to three conditions: stabilizability in the behavioral sense, detectability in the behavioral sense, and the condition that all transmission zeros of the system are in the open left complex half-plane. Furthermore, for the same class, it is shown that we can achieve, by a compatible control in the behavioral sense, that the Lyapunov exponent of the interconnected system equals the Lyapunov exponent of the zero dynamics.

Keywords

Differential-algebraic equations Zero dynamics Transmission zeros Right-invertibility Stabilizability Detectability Lyapunov exponent 

Notes

Acknowledgements

I am indebted to Achim Ilchmann (Ilmenau University of Technology) for several constructive discussions.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. No. 65 in Pure and Applied Mathematics. Academic, New York/London (1975)Google Scholar
  2. 2.
    Bender, D., Laub, A.: The linear quadratic optimal regulator problem for descriptor systems. IEEE Trans. Autom. Control 32, 672–688 (1987)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Berger, T.: Bohl exponent for time-varying linear differential-algebraic equations. Int. J. Control 85(10), 1433–1451 (2012)CrossRefMATHGoogle Scholar
  4. 4.
    Berger, T.: On differential-algebraic control systems. Ph.D. thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Ilmenau (2013)Google Scholar
  5. 5.
    Berger, T.: Zero dynamics and funnel control of general linear differential-algebraic systems (2013). Submitted for publication. Preprint available from the website of the authorGoogle Scholar
  6. 6.
    Berger, T., Ilchmann, A., Reis, T.: Normal forms, high-gain, and funnel control for linear differential-algebraic systems. In: Biegler, L.T., Campbell, S.L., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints. Advances in Design and Control, vol. 23, pp. 127–164. SIAM, Philadelphia (2012)CrossRefGoogle Scholar
  7. 7.
    Berger, T., Ilchmann, A., Reis, T.: Zero dynamics and funnel control of linear differential-algebraic systems. Math. Control Signals Syst. 24(3), 219–263 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Berger, T., Reis, T.: Controllability of linear differential-algebraic systems – a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, pp. 1–61. Springer, Berlin/Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Berger, T., Trenn, S.: The quasi-Kronecker form for matrix pencils. SIAM J. Matrix Anal. Appl. 33(2), 336–368 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Berger, T., Trenn, S.: Addition to “The quasi-Kronecker form for matrix pencils”. SIAM J. Matrix Anal. Appl. 34(1), 94–101 (2013)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N.K.: Feedback design for regularizing descriptor systems. Linear Algebra Appl. 299, 119–151 (1999)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Campbell, S.L., Kunkel, P., Mehrmann, V.: Regularization of linear and nonlinear descriptor systems. In: Biegler, L.T., Campbell, S.L., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints. Advances in Design and Control, vol. 23, pp. 17–36. SIAM, Philadelphia (2012)CrossRefGoogle Scholar
  13. 13.
    Francis, B.A., Wonham, W.M.: The role of transmission zeros in linear multivariable regulators. Int. J. Control 22(5), 657–681 (1975)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Hou, M., Müller, P.C.: Observer design for descriptor systems. IEEE Trans. Autom. Control 44(1), 164–168 (1999)CrossRefMATHGoogle Scholar
  15. 15.
    Isidori, A.: Nonlinear Control Systems. Communications and Control Engineering Series, 3rd edn. Springer, Berlin (1995)Google Scholar
  16. 16.
    Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs (1980)MATHGoogle Scholar
  17. 17.
    Lewis, F.L.: A survey of linear singular systems. IEEE Proc. Circuits Syst. Signal Process. 5(1), 3–36 (1986)CrossRefMATHGoogle Scholar
  18. 18.
    Linh, V.H., Mehrmann, V.: Lyapunov, Bohl and Sacker-Sell spectral intervals for differential-algebraic equations. J. Dyn. Differ. Equ. 21, 153–194 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Loiseau, J.J., Özçaldiran, K., Malabre, M., Karcanias, N.: Feedback canonical forms of singular systems. Kybernetika 27(4), 289–305 (1991)MATHMathSciNetGoogle Scholar
  20. 20.
    Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory. A Behavioral Approach. Springer, New York (1998)CrossRefGoogle Scholar
  21. 21.
    Rosenbrock, H.H.: State Space and Multivariable Theory. Wiley, New York (1970)MATHGoogle Scholar
  22. 22.
    Trentelman, H.L., Stoorvogel, A.A., Hautus, M.L.J.: Control Theory for Linear Systems. Communications and Control Engineering. Springer, London (2001)CrossRefMATHGoogle Scholar
  23. 23.
    Varga, A.: On stabilization methods of descriptor systems. Syst. Control Lett. 24, 133–138 (1995)CrossRefMATHGoogle Scholar
  24. 24.
    Willems, J.C.: The behavioral approach to open and interconnected systems. IEEE Control Syst. Mag. 27(6), 46–99 (2007)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Xu, S., Van Dooren, P., Ştefan, R., Lam, J.: Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Trans. Autom. Control 47(7), 1122–1128 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität HamburgHamburgGermany

Personalised recommendations